Structural Engineering and Mechanics

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Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation

  • Karami, Behrouz (Department of Mechanical Engineering, Marvdasht Branch, Islamic Azad University) ;
  • Janghorban, Maziar (Department of Mechanical Engineering, Marvdasht Branch, Islamic Azad University) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department)
  • Received : 2018.06.15
  • Accepted : 2019.02.12
  • Published : 2019.04.10
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Abstract

This work deals with the size-dependent wave propagation analysis of functionally graded (FG) anisotropic nanoplates based on a nonlocal strain gradient refined plate model. The present model incorporates two scale coefficients to examine wave dispersion relations more accurately. Material properties of FG anisotropic nanoplates are exponentially varying in the z-direction. In order to solve the governing equations for bulk waves, an analytical method is performed and wave frequencies and phase velocities are obtained as a function of wave number. The influences of several important parameters such as material graduation exponent, geometry, Winkler-Pasternak foundation parameters and wave number on the wave propagation of FG anisotropic nanoplates resting on the elastic foundation are investigated and discussed in detail. It is concluded that these parameters play significant roles on the wave propagation behavior of the nanoplates. From the best knowledge of authors, it is the first time that FG nanoplate made of anisotropic materials is investigated, so, presented numerical results can serve as benchmarks for future analysis of such structures.

Keywords

References

  1. Aifantis, E.C. (1992), "On the role of gradients in the localization of deformation and fracture", Int. J. Eng. Sci., 30(10), 1279-1299. https://doi.org/10.1016/0020-7225(92)90141-3
  2. Aminipour, H. and Janghorban, M. (2017), "Wave propagation in anisotropic plates using trigonometric shear deformation theory", Mech. Adv. Mater. Struct., 24(13), 1135-1144. https://doi.org/10.1080/15376494.2016.1227500
  3. Aminipour, H., Janghorban, M. and Li, L. (2018), "A new model for wave propagation in functionally graded anisotropic doublycurved shells", Compos. Struct., 190, 91-111. https://doi.org/10.1016/j.compstruct.2018.02.003
  4. Ansari, R., Rouhi, H. and Sahmani, S. (2011), "Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics", Int. J. Mech. Sci., 53(9), 786-792. https://doi.org/10.1016/j.ijmecsci.2011.06.010
  5. Apuzzo, A., Barretta, R., Luciano, R., De Sciarra, F.M. and Penna, R. (2017), "Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model", Compos. Part B: Eng., 123, 105-111. https://doi.org/10.1016/j.compositesb.2017.03.057
  6. Barati, M.R. (2017), "Vibration analysis of FG nanoplates with nanovoids on viscoelastic substrate under hygro-thermomechanical loading using nonlocal strain gradient theory", Struct. Eng. Mech., 64(6), 683-693. https://doi.org/10.12989/SEM.2017.64.6.683
  7. Barretta, R., Luciano, R., De Sciarra, F.M. and Ruta, G. (2018), "Stress-driven nonlocal integral model for Timoshenko elastic nano-beams", Eur. J. Mech.-A/Sol., 72, 275-286.
  8. Batra, R., Qian, L. and Chen, L. (2004), "Natural frequencies of thick square plates made of orthotropic, trigonal, monoclinic, hexagonal and triclinic materials", J. Sound Vibr., 270(4-5), 1074-1086. https://doi.org/10.1016/S0022-460X(03)00625-4
  9. Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S. (2017), "A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams", Struct. Eng. Mech., 62(6), 695-702. https://doi.org/10.12989/SEM.2017.62.6.695
  10. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  11. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., 18(2), 425-442. https://doi.org/10.12989/SCS.2015.18.2.425
  12. Challamel, N. and Wang, C. (2008), "The small length scale effect for a non-local cantilever beam: A paradox solved", Nanotechnol., 19(34), 345703. https://doi.org/10.1088/0957-4484/19/34/345703
  13. Dash, S., Mehar, K., Sharma, N., Mahapatra, T.R. and Panda, S.K. (2018a), "Modal analysis of FG sandwich doubly curved shell structure", Struct. Eng. Mech., 68(6), 721-733. https://doi.org/10.12989/sem.2018.68.6.721
  14. Dash, S., Sharma, N., Mahapatra, T., Panda, S. and Sahu, P. (2018b), "Free vibration analysis of functionally graded sandwich flat panel", Mater. Sci. Eng., 377(1).
  15. Dutta, G., Panda, S.K., Mahapatra, T.R. and Singh, V.K. (2017), "Electro-magneto-elastic response of laminated composite plate: A finite element approach", Int. J. Appl. Comput. Math., 3(3), 2573-2592. https://doi.org/10.1007/s40819-016-0256-6
  16. Ebrahimi, F. and Barati, M.R. (2016), "A nonlocal higher-order shear deformation beam theory for vibration analysis of sizedependent functionally graded nanobeams", Arab. J. Sci. Eng., 41(5), 1679-1690. https://doi.org/10.1007/s13369-015-1930-4
  17. Ehyaei, J., Farazmandnia, N. and Jafari, A. (2017), "Rotating effects on hygro-mechanical vibration analysis of FG beams based on Euler-Bernoulli beam theory", Struct. Eng. Mech., 63(4), 471-480. https://doi.org/10.12989/SEM.2017.63.4.471
  18. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  19. Farokhi, H. and Ghayesh, M.H. (2015), "Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory", Int. J. Mech. Sci., 90, 133-144. https://doi.org/10.1016/j.ijmecsci.2014.11.002
  20. Gholipour, A., Farokhi, H. and Ghayesh, M.H. (2015), "In-plane and out-of-plane nonlinear size-dependent dynamics of microplates", Nonlin. Dyn., 79(3), 1771-1785. https://doi.org/10.1007/s11071-014-1773-7
  21. Guo, J., Chen, J. and Pan, E. (2017), "Free vibration of threedimensional anisotropic layered composite nanoplates based on modified couple-stress theory", Phys. E: Low-Dimens. Syst. Nanostruct., 87, 98-106. https://doi.org/10.1016/j.physe.2016.11.025
  22. Hirwani, C., Biswash, S., Mehar, K. and Panda, S.K. (2018), "Numerical flexural strength analysis of thermally stressed delaminated composite structure under sinusoidal loading", Mater. Sci. Eng., 012019.
  23. Hirwani, C.K. and Panda, S.K. (2018), "Numerical and experimental validation of nonlinear deflection and stress responses of pre-damaged glass-fibre reinforced composite structure", Ocean Eng., 159, 237-252. https://doi.org/10.1016/j.oceaneng.2018.04.035
  24. Hirwani, C.K. and Panda, S.K. (2019), "Nonlinear finite element solutions of thermoelastic deflection and stress responses of internally damaged curved panel structure", Appl. Math. Modell., 65, 303-317. https://doi.org/10.1016/j.apm.2018.08.014
  25. Karami, B. and Janghorban, M. (2016), "Effect of magnetic field on the wave propagation in nanoplates based on strain gradient theory with one parameter and two-variable refined plate theory", Mod. Phys. Lett. B, 30(36), 1650421. https://doi.org/10.1142/S0217984916504212
  26. Karami, B. and Janghorban, M. (2019), "On the dynamics of porous nanotubes with variable material properties and variable thickness", Int. J. Eng. Sci., 136, 53-66. https://doi.org/10.1016/j.ijengsci.2019.01.002
  27. Karami, B., Janghorban, M. and Li, L. (2018a), "On guided wave propagation in fully clamped porous functionally graded nanoplates", Acta Astronaut., 143, 380-390. https://doi.org/10.1016/j.actaastro.2017.12.011
  28. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018b), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28(1), 99-110. https://doi.org/10.12989/SCS.2018.28.1.099
  29. Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., 25(3), 361-374. https://doi.org/10.12989/SCS.2017.25.3.361
  30. Karami, B., Janghorban, M. and Tounsi, A. (2018c), "Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions", Eng. Comput., 1-20.
  31. Karami, B., Janghorban, M. and Tounsi, A. (2018d), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., 27(2), 201-216. https://doi.org/10.12989/SCS.2018.27.2.201
  32. Karami, B., Janghorban, M. and Tounsi, A. (2018e), "Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory", Thin-Wall. Struct., 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025
  33. Karami, B. and Karami, S. (2019), "Buckling analysis of nanoplate-type temperature-dependent heterogeneous materials", Adv. Nano Res., 7(1), 51-61. https://doi.org/10.12989/ANR.2019.7.1.051
  34. Karami, B., Shahsavari, D. and Janghorban, M. (2018f), "A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates", Aerosp. Sci. Technol., 82, 499-512. https://doi.org/10.1016/j.ast.2018.10.001
  35. Karami, B., Shahsavari, D. and Janghorban, M. (2018g), "Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory", Mech. Adv. Mater. Struct., 25(12), 1047-1057. https://doi.org/10.1080/15376494.2017.1323143
  36. Karami, B., Shahsavari, D., Janghorban, M., Dimitri, R. and Tornabene, F. (2019a), "Wave propagation of porous nanoshells", Nanomater., 9(1), 22. https://doi.org/10.3390/nano9010022
  37. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2018h), "Wave dispersion of mounted graphene with initial stress", Thin-Wall. Struct., 122, 102-111. https://doi.org/10.1016/j.tws.2017.10.004
  38. Karami, B., Shahsavari, D., Karami, M. and Li, L. (2019), "Hygrothermal wave characteristic of nanobeam-type inhomogeneous materials with porosity under magnetic field", J. Mech. Eng. Sci., 233(6), 2149-2169. https://doi.org/10.1177/0954406218781680
  39. Karami, B., Shahsavari, D. and Li, L. (2018j), "Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory", Phys. E: Low-Dimens. Syst. Nanostruct., 97, 317-327. https://doi.org/10.1016/j.physe.2017.11.020
  40. Karami, B., Shahsavari, D. and Li, L. (2018k), "Temperaturedependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field", J. Therm. Stress., 41(4), 483-499. https://doi.org/10.1080/01495739.2017.1393781
  41. Karami, B., Shahsavari, D., Li, L., Karami, M. and Janghorban, M. (2019b), "Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory", J. Mech. Eng. Sci., 233(1), 287-301. https://doi.org/10.1177/0954406218756451
  42. Karami, B., Shahsavari, D., Nazemosadat, S.M.R., Li, L. and Ebrahimi, A. (2018l), "Thermal buckling of smart porous functionally graded nanobeam rested on Kerr foundation", Steel Compos. Struct., 29(3), 349-362. https://doi.org/10.12989/SCS.2018.29.3.349
  43. Katariya, P.V., Hirwani, C.K. and Panda, S.K. (2018), "Geometrically nonlinear deflection and stress analysis of skew sandwich shell panel using higher-order theory", Eng. Comput., 1-19.
  44. Katariya, P.V., Panda, S.K., Hirwani, C.K., Mehar, K. and Thakare, O. (2017a), "Enhancement of thermal buckling strength of laminated sandwich composite panel structure embedded with shape memory alloy fibre", Smart Struct. Syst., 20(5), 595-605. https://doi.org/10.12989/SSS.2017.20.5.595
  45. Katariya, P.V., Panda, S.K. and Mahapatra, T.R. (2017b), "Nonlinear thermal buckling behaviour of laminated composite panel structure including the stretching effect and higher-order finite element", Adv. Mater. Res., 6(4), 349-361. https://doi.org/10.12989/AMR.2017.6.4.349
  46. Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2017), "A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates", Struct. Eng. Mech., 64(4), 391-402. https://doi.org/10.12989/SEM.2017.64.4.391
  47. Lagnese, J. (1989), Boundary Stabilization of Thin Plates, Philadelphia, SIAM Studies in Applied Mathematics, 10.
  48. Lam, D.C., Yang, F., Chong, A., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Sol., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  49. Li, L. and Hu, Y. (2015), "Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory", Int. J. Eng. Sci., 97, 84-94. https://doi.org/10.1016/j.ijengsci.2015.08.013
  50. Li, L., Hu, Y. and Ling, L. (2016), "Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory", Phys. E: Low-Dimes. Syst. Nanostruct., 75, 118-124. https://doi.org/10.1016/j.physe.2015.09.028
  51. Lim, C., Zhang, G. and Reddy, J. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Sol., 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001
  52. Ma, H., Gao, X.L. and Reddy, J. (2008), "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Sol., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  53. Mahapatra, T.R., Mehar, K., Panda, S.K., Dewangan, S. and Dash, S. (2017), "Flexural strength of functionally graded nanotube reinforced sandwich spherical panel", Mater. Sci. Eng., 178(1), 012031.
  54. Mehar, K. and Panda, S.K. (2017a), "Numerical investigation of nonlinear thermomechanical deflection of functionally graded CNT reinforced doubly curved composite shell panel under different mechanical loads", Compos. Struct., 161, 287-298. https://doi.org/10.1016/j.compstruct.2016.10.135
  55. Mehar, K. and Panda, S.K. (2017b), "Thermoelastic analysis of FG-CNT reinforced shear deformable composite plate under various loadings", Int. J. Comput. Meth., 14(2), 1750019. https://doi.org/10.1142/S0219876217500190
  56. Mehar, K. and Panda, S.K. (2018), "Thermoelastic flexural analysis of FG-CNT doubly curved shell panel", Aircraft Eng. Aerosp. Technol., 90(1), 11-23. https://doi.org/10.1108/AEAT-11-2015-0237
  57. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018a), "Large deformation bending responses of nanotube-reinforced polymer composite panel structure: Numerical and experimental analyses", J. Aerosp. Eng., 0954410018761.
  58. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018b), "Thermoelastic deflection responses of CNT reinforced sandwich shell structure using finite element method", Sci. Iranic., 25(5), 2722-2737.
  59. Mehar, K., Panda, S.K. and Patle, B.K. (2017), "Thermoelastic vibration and flexural behavior of FG-CNT reinforced composite curved panel", Int. J. Appl. Mech., 9(4), 1750046. https://doi.org/10.1142/S1758825117500466
  60. Mehar, K., Panda, S.K. and Patle, B.K. (2018c), "Stress, deflection, and frequency analysis of CNT reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach", Polym. Compos., 39(10), 3792-3809. https://doi.org/10.1002/pc.24409
  61. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Rat. Mech. Analy., 16(1), 51-78. https://doi.org/10.1007/BF00248490
  62. Mirjavadi, S.S., Afshari, B.M., Shafiei, N., Hamouda, A. and Kazemi, M. (2017), "Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams", Steel Compos. Struct., 25(4), 415-426. https://doi.org/10.12989/SCS.2017.25.4.415
  63. Moradweysi, P., Ansari, R., Hosseini, K. and Sadeghi, F. (2018), "Application of modified Adomian decomposition method to pull-in instability of nano-switches using nonlocal Timoshenko beam theory", Appl. Math. Modell., 54, 594-604. https://doi.org/10.1016/j.apm.2017.10.011
  64. Nami, M.R. and Janghorban, M. (2014), "Wave propagation in rectangular nanoplates based on strain gradient theory with one gradient parameter with considering initial stress", Mod. Phys. Lett. B, 28(3), 1450021. https://doi.org/10.1142/S0217984914500213
  65. Nami, M.R. and Janghorban, M. (2015), "Free vibration analysis of rectangular nanoplates based on two-variable refined plate theory using a new strain gradient elasticity theory", J. Brazil. Soc. Mech. Sci. Eng., 37(1), 313-324. https://doi.org/10.1007/s40430-014-0169-4
  66. Nejad, M.Z. and Hadi, A. (2016), "Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 106, 1-9. https://doi.org/10.1016/j.ijengsci.2016.05.005
  67. Pan, E. (2003), "Exact solution for functionally graded anisotropic elastic composite laminates", J. Compos. Mater., 37(21), 1903-1920. https://doi.org/10.1177/002199803035565
  68. Polizzotto, C. (2003), "Gradient elasticity and nonstandard boundary conditions", Int. J. Sol. Struct., 40(26), 7399-7423. https://doi.org/10.1016/j.ijsolstr.2003.06.001
  69. Rahmani, O., Refaeinejad, V. and Hosseini, S. (2017), "Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams", Steel Compos. Struct., 23(3), 339-350. https://doi.org/10.12989/scs.2017.23.3.339
  70. Romano, G. and Barretta, R. (2017), "Nonlocal elasticity in nanobeams: The stress-driven integral model", Int. J. Eng. Sci., 115, 14-27. https://doi.org/10.1016/j.ijengsci.2017.03.002
  71. Romano, G., Barretta, R. and Diaco, M. (2017), "On nonlocal integral models for elastic nano-beams", Int. J. Mech. Sci., 131, 490-499. https://doi.org/10.1016/j.ijmecsci.2017.07.013
  72. Sahmani, S., Aghdam, M.M. and Rabczuk, T. (2018a), "Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory", Compos. Struct., 186, 68-78. https://doi.org/10.1016/j.compstruct.2017.11.082
  73. Sahmani, S., Aghdam, M.M. and Rabczuk, T. (2018b), "Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs", Compos. Struct., 198, 51-62. https://doi.org/10.1016/j.compstruct.2018.05.031
  74. Sahmani, S. and Fattahi, A. (2017), "Calibration of developed nonlocal anisotropic shear deformable plate model for uniaxial instability of 3D metallic carbon nanosheets using MD simulations", Comput. Meth. Appl. Mech. Eng., 322, 187-207. https://doi.org/10.1016/j.cma.2017.04.015
  75. Sahoo, S.S., Panda, S.K. and Singh, V.K. (2017), "Experimental and numerical investigation of static and free vibration responses of woven glass/epoxy laminated composite plate", J. Mater.: Des. Appl., 231(5), 463-478.
  76. Sayyad, A.S. and Ghugal, Y.M. (2018), "An inverse hyperbolic theory for FG beams resting on Winkler-Pasternak elastic foundation", Adv. Aircr. Spacecr. Sci., 5(6), 671-689. https://doi.org/10.12989/AAS.2018.5.6.671
  77. Shafiei, N. and She, G.L. (2018), "On vibration of functionally graded nano-tubes in the thermal environment", Int. J. Eng. Sci., 133, 84-98. https://doi.org/10.1016/j.ijengsci.2018.08.004
  78. Shahsavari, D. and Janghorban, M. (2017), "Bending and shearing responses for dynamic analysis of single-layer graphene sheets under moving load", J. Brazil. Soc. Mech. Sci. Eng., 39(10), 3849-3861. https://doi.org/10.1007/s40430-017-0863-0
  79. Shahsavari, D., Karami, B., Fahham, H.R. and Li, L. (2018a), "On the shear buckling of porous nanoplates using a new sizedependent quasi-3D shear deformation theory", Acta Mech., 229(11), 4549-4573. https://doi.org/10.1007/s00707-018-2247-7
  80. Shahsavari, D., Karami, B., Janghorban, M. and Li, L. (2017), "Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment", Mater. Res. Expr., 4(8), 085013. https://doi.org/10.1088/2053-1591/aa7d89
  81. Shahsavari, D., Karami, B. and Li, L. (2018b), "Damped vibration of a graphene sheet using a higher-order nonlocal straingradient Kirchhoff plate model", Compt. Rend. Mecan., 346(12), 1216-1232. https://doi.org/10.1016/j.crme.2018.08.011
  82. Shahsavari, D., Karami, B. and Li, L. (2018c), "A high-order gradient model for wave propagation analysis of porous FG nanoplates", Steel Compos. Struct., 29(1), 53-66. https://doi.org/10.12989/scs.2018.29.1.053
  83. Shahsavari, D., Karami, B. and Mansouri, S. (2018d), "Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories", Eur. J. Mech.-A/Sol., 67, 200-214. https://doi.org/10.1016/j.euromechsol.2017.09.004
  84. Shahsavari, D., Shahsavari, M., Li, L. and Karami, B. (2018e), "A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation", Aerosp. Sci. Technol., 72, 134-149. https://doi.org/10.1016/j.ast.2017.11.004
  85. She, G.L., Ren, Y.R., Yuan, F.G. and Xiao, W.S. (2018a), "On vibrations of porous nanotubes", Int. J. Eng. Sci., 125, 23-35. https://doi.org/10.1016/j.ijengsci.2017.12.009
  86. She, G.L., Yan, K.M., Zhang, Y.L., Liu, H.B. and Ren, Y.R. (2018b), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Eur. Phys. J. Plus, 133(9), 368. https://doi.org/10.1140/epjp/i2018-12196-5
  87. She, G.L., Yuan, F.G., Karami, B., Ren, Y.R. and Xiao, W.S. (2019), "On nonlinear bending behavior of FG porous curved nanotubes", Int. J. Eng. Sci., 135, 58-74. https://doi.org/10.1016/j.ijengsci.2018.11.005
  88. She, G.L., Yuan, F.G. and Ren, Y.R. (2018c), "On wave propagation of porous nanotubes", Int. J. Eng. Sci., 130, 62-74. https://doi.org/10.1016/j.ijengsci.2018.05.002
  89. She, G.L., Yuan, F.G., Ren, Y.R. and Xiao, W.S. (2017), "On buckling and postbuckling behavior of nanotubes", Int. J. Eng. Sci., 121, 130-142. https://doi.org/10.1016/j.ijengsci.2017.09.005
  90. Shimpi, R. and Patel, H. (2006), "Free vibrations of plate using two variable refined plate theory", J. Sound Vibr., 296(4), 979-999. https://doi.org/10.1016/j.jsv.2006.03.030
  91. Shimpi, R.P. (2002), "Refined plate theory and its variants", AIAA J., 40(1), 137-146. https://doi.org/10.2514/2.1622
  92. Simsek, M. (2011), "Forced vibration of an embedded singlewalled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel Compos. Struct., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059
  93. Singh, V.K. and Panda, S.K. (2017), "Geometrical nonlinear free vibration analysis of laminated composite doubly curved shell panels embedded with piezoelectric layers", J. Vibr. Contr., 23(13), 2078-2093. https://doi.org/10.1177/1077546315609988
  94. Sobhy, M. (2017), "Hygro-thermo-mechanical vibration and buckling of exponentially graded nanoplates resting on elastic foundations via nonlocal elasticity theory", Struct. Eng. Mech., 63(3), 401-415. https://doi.org/10.12989/SEM.2017.63.3.401
  95. Volokh, K.Y. and Hutchinson, J. (2002), "Are lower-order gradient theories of plasticity really lower order?", J. Appl. Mech., 69(6), 862-864. https://doi.org/10.1115/1.1504096
  96. Xiao, W., Li, L. and Wang, M. (2017), "Propagation of in-plane wave in viscoelastic monolayer graphene via nonlocal strain gradient theory", Appl. Phys. A, 123(6), 388. https://doi.org/10.1007/s00339-017-1007-1
  97. Yang, F., Chong, A., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", International J. Sol. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  98. Zhu, X. and Li, L. (2017), "Closed form solution for a nonlocal strain gradient rod in tension", Int. J. Eng. Sci., 119, 16-28. https://doi.org/10.1016/j.ijengsci.2017.06.019

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